| dc.description.abstract | This thesis concerns the concept of metamathematical fixed points. After an
introduction, we survey the field of metamathematics, from la fin du siècle to
present. We are especially interested in the notion of fixed points, theorems on
the existence of various kinds of fixed points, and their applications to metamathematics. The second part of the thesis is a technical investigation of sets
of fixed points. Given some recursively enumerable, consistent extension T of
Peano arithmetic, we define for each formula θ(x) the set
Fix^T (θ) := {δ : T |- δ ↔ θ(δ)}.
Our main result on these sets is that they are all Σ_1-complete. Furthermore,
we define for each formula θ(x), the set
Fix_Γ^T (θ) := {δ : T |- δ ↔ θ(δ)},
where δ is a sentence in Γ. Using methods of Bennet, Bernardi, Guaspari,
Lindström, and Smullyan, we characterise these sets for formulas in Γ' ⊃ Γ,
and provide partial results for formulas in Γ. We give a sufficient condition on
recursive sets to be a set of fixed points, and show that such sets exists. We also
present a sufficient condition for a recursively enumerable set of Γ-sentences to
be a set of fixed points of a Γ-formula.
In the following section, we study the structure of sets of fixed points ordered
under set inclusion, and prove certain properties on these structures. Finally, we
connect our research to another open problem of metamathematics, and state
some possible further work. | sv |
